Expected regions cut by two random chords

Question

Choose four points $X_1,X_2,X_3,X_4$ independently and uniformly on a circle. Draw the chord $X_1X_2$ and the chord $X_3X_4$ . What is the expected number of regions into which the disc is divided?

Short answer

$10/3$ . Two chords give $3$ regions if they don't cross and $4$ if they do. They cross exactly when the endpoints alternate around the circle, which happens in $1$ of the $3$ equally likely pairings, so the expectation is $3+\tfrac13=\tfrac{10}{3}$ .

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Expectation, Mean, and Variance
Independence
Conditional Expectation and Variance Revisited

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